cs685-rbm - Matrices Linear Algebra Review Rigid Body...

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1 • Linear Algebra Review Rigid Body Motion in 2D Rigid Body Motion in 3D Jana Kosecka, CS 685 Why do we need Linear Algebra? We will associate coordinates to – 3D points in the scene – 2D points in the CCD array – 2D points in the image Coordinates will be used to – Perform geometrical transformations – Associate 3D with 2D points Images are matrices of numbers – We will find properties of these numbers 2 Matrices m n m n m n B A C ! ! ! + = ij ij ij b a c + = ! " # $ % = ! " # $ % + ! " # $ % 6 4 7 8 5 1 2 6 1 3 5 2 Matrices p m m n p n B A C ! ! ! = ! = = m k kj ik ij b a c 1 ! " # $ % = ! " # $ % ! " # $ % 11 19 29 17 5 1 2 6 . 1 3 5 2 ! " # $ % = ! " # $ % ! " # $ % 10 17 32 18 1 3 5 2 . 5 1 2 6 n n n n n n n n A B B A ! ! ! ! "
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3 Matrices m n T n m A C ! ! = ji ij a c = T T T A B AB = ) ( T T T B A B A + = + ) ( A A T = ! " # $ % = ! " # $ % 5 2 1 6 5 1 2 6 T ! " # $ % = ! ! ! " # $ $ $ % 8 5 2 3 1 6 8 3 5 1 2 6 T Matrices 13 15 2 1 3 5 2 det ! = ! = " # $ % 32 31 22 21 13 33 31 23 21 12 33 32 23 22 11 33 32 31 23 22 21 13 12 11 det a a a a a a a a a a a a
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cs685-rbm - Matrices Linear Algebra Review Rigid Body...

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